# Missing exponent in "Theoretical Minimum"?

• I
Gold Member

## Summary:

Is there a missing exponent in the authors' application of the Euler-Lagrange equation?
In "The Theoretical Minimum" (the one on classical mechanics), on page 218, the authors write a Lagrangian

$$L=\frac m 2 (\dot r^2 +r^2\dot \theta ^2)+\frac {GMm} r$$

They then apply the Euler-Lagrange equation ##\frac d {dt}\frac {dL} {d\dot r}=\frac {dL} {dr}## (I know there should be partial derivatives there but I couldn't find the symbol for it in my Latex primer; if anyone could enlighten me, I'd appreciate it) and wrote the result...

$$\ddot r=r\dot \theta^2-\frac {GM} r$$

My question is, shouldn't the last r in the denominator be squared since it results from differentiating the GMm/r term in the Lagrangian by r?

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vanhees71
Gold Member
2019 Award
Indeed, there should be an ##r^2## in the denominator.

dextercioby and Ibix
Ibix
...and \partial gives you ##\partial##.

SamRoss and vanhees71
Gold Member
Indeed, there should be an ##r^2## in the denominator.
Thanks, I thought so. The original version of the book apparently lacked both exponents in the result. The online errata for the book shows that, as a correction, the first exponent (above theta dot) was put in but the second exponent is still missing. Weird. However, on the next page of the book the entire equation is written out again, correctly this time. I hadn't moved on to that page before making this post.

vanhees71